4. (a) Let I = [a, b] and let f and g be bounded functions on I to R. Suppose that f(x) < g(x) for all x E I. Show that L(f) < L(g) and U(f)
4. (a) Let I = [a, b] and let f and g be bounded functions on I to R. Suppose that f(x) < g(x) for all x E I. Show that L(f) < L(g) and U(f)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
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![4.
(a) Let I = [a, b] and let f and g be bounded functions on I to R. Suppose
that f(x) < g(x) for all x E I. Show that L(f) < L(g) and U(f) <U(g).
(b) Let I = [a, b] and let f, g, h be bounded functions on I to R. Suppose
that f(x) < g(x) < h(x) for x E I. Show that if f and h are integrable on I
and if
A
| f(x)dr = | h(x)dx,
then
is integrable on I and " g(x)dx
А.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5d456ce1-cbfb-470f-8ef9-05bd7d57f044%2Fb308b460-4aa4-4d15-a56e-e5dba62f5061%2F3r0nj2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.
(a) Let I = [a, b] and let f and g be bounded functions on I to R. Suppose
that f(x) < g(x) for all x E I. Show that L(f) < L(g) and U(f) <U(g).
(b) Let I = [a, b] and let f, g, h be bounded functions on I to R. Suppose
that f(x) < g(x) < h(x) for x E I. Show that if f and h are integrable on I
and if
A
| f(x)dr = | h(x)dx,
then
is integrable on I and " g(x)dx
А.
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